{
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   "source": [
    "# The Convex Hull Problem"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "Pound a bunch of nails into a board, then stretch a rubber band around them and let the rubber band snap taut, like this:\n",
    "\n",
    "<img src=\"convexhull.jpg\">\n",
    "\n",
    "The rubber band has traced out the *convex hull* of the set of nails. It turns out this is an important problem with applications in computer graphics, robot motion planning, geographical information systems, ethology, and other areas.\n",
    "More formally, we say that:\n",
    "\n",
    "*Given a finite set, **P**, of points in a plane, the convex hull of **P** is a polygon, **H**, such that:*\n",
    "\n",
    "- *Every point in **P** lies either on or inside of **H**.*\n",
    "- *Every vertex of **H** is a point in **P**.*\n",
    "- **H** *is convex: a line segment joining any two vertexes of **H** either is an edge of **H** or lies inside **H**.*\n",
    "\n",
    "\n",
    "In this notebook we develop an algorithm to find the convex hull (and show examples of how to use `matplotlib` plotting). The first thing to do is decide how we will represent the objects of interest:\n",
    "\n",
    "- **Point**: We'll define a class such that `Point(3, 4)` is a point where `p.x` is 3 and `p.y` is 4.\n",
    "- **Set of Points**: We'll use a Python set: `{Point(0,0), Point(3,4), ...}`\n",
    "- **Polygon**: We'll represent a polygon as an ordered list of vertex points.\n",
    "\n",
    "First, get the necessary imports done:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
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   "source": [
    "from __future__ import division, print_function\n",
    "\n",
    "%matplotlib inline \n",
    "import matplotlib.pyplot as plt\n",
    "import collections\n",
    "import random\n",
    "import math"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "# Points and Sets of Points\n",
    "\n",
    "I'll define the class `Point` as a named tuple of `x` and `y` coordinates, and `Points(n)` as a function that creates a set of *n* random points. \n",
    "\n",
    "There are two complications to the function `Points(n)`:\n",
    "1. A second optional argument is used to set the random seed.  This way, the same call to `Points` will return the same result each time.  That makes it easier to reproduce tests.  If you want different sets of points, just pass in different values for the seed.\n",
    "2. Since `matplotlib` plots on a 3&times;2 rectangle by default, the points will be uniformly sampled   from a 3&times;2 box (with a small border of 0.05 on each edge to prevent the points from bumping up against the edge of the box)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
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   "outputs": [],
   "source": [
    "Point = collections.namedtuple('Point', 'x, y')\n",
    "\n",
    "def Points(n, seed=42):\n",
    "    \"Generate n random points within a 3 x 2 box.\"\n",
    "    random.seed((n, seed))\n",
    "    b = 0.05 # border\n",
    "    return {Point(random.uniform(b, 3-b), random.uniform(b, 2-b)) \n",
    "            for _ in range(n)}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
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    "collapsed": false,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "{Point(x=0.15172583449638682, y=1.6108693392839208),\n",
       " Point(x=0.968326330695687, y=1.3139550880088586),\n",
       " Point(x=1.3508070075242857, y=0.22290610532132638)}"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Points(3)"
   ]
  },
  {
   "cell_type": "markdown",
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    }
   },
   "source": [
    "# Visualizing Points and Line Segments\n",
    "\n",
    "\n",
    "Now let's see how to visualize points; I'll define a function `plot_points`.  We will want to be able to see:\n",
    "- The **points** themselves. \n",
    "- Optionally, **line segments** between points. An optional `style` parameter allows you to specify whether you want lines or not, and what color they should be. This parameter uses the standard [style format](http://matplotlib.org/1.3.1/api/pyplot_api.html#matplotlib.pyplot.plot) defined by matplotlib; for example, `'r.'` means red colored dots with no lines, `'bs-'` means blue colored squares with lines between them, and `'go:'` means green colored circles with dotted lines between them.  The lines go from point to point in order; if you want the lines to close\n",
    "back from the last point to the first (to form a complete polygon), specify `closed=True`. (For that to work,\n",
    "the collection of points must be a list; with `closed=False` the collection can be any collection.)\n",
    "- Optionally, **labels** on the points that let us distinguish one from another. You get\n",
    "labels (integers from 0 to *n*) if you specify `labels=True`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
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   "outputs": [],
   "source": [
    "def plot_points(points, style='r.', labels=False, closed=False): \n",
    "    \"\"\"Plot a collection of points. Optionally change the line style, label points with numbers, \n",
    "    and/or form a closed polygon by closing the line from the last point to the first.\"\"\"\n",
    "    if labels:\n",
    "        for (i, (x, y)) in enumerate(points):\n",
    "            plt.text(x, y, '  '+str(i))\n",
    "    if closed:\n",
    "        points = points + [points[0]]\n",
    "    plt.plot([p.x for p in points], [p.y for p in points], style, linewidth=2.5)\n",
    "    plt.axis('scaled'); plt.axis('off')"
   ]
  },
  {
   "cell_type": "markdown",
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   },
   "source": [
    "Here's an example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
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   "outputs": [
    {
     "data": {
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4d36u/2Bm/0j7fTpdIfoOeYdLpxCOA3RBmEbFzfLA6T6D35BX21wD4EftBNHhTc9zTf5+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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c44dda0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_points(Points(200))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "# Convexity\n",
    "\n",
    "\n",
    "We want to make a *convex* hull, so we better have some way of determining whether a polygon is *convex*. Let's examine one that is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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EKNk6uKpTYaKaWGKTeh1c1akwKakmlpgkXQdXdeYAtkc1scQihzq4qnM7k0FqYjP7hJmt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\niAWdyUSNAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c44dcf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "octagon = [Point(-10, 0), Point(-7, -7), Point(0, -10), Point(+7, -7), \n",
    "           Point(+10, 0), Point(+7, +7), Point(0, +10), Point(-7, 7)]\n",
    "plot_points(octagon, 'bs-', labels=True, closed=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "If you start at point 0 at the left and proceed in order counterclockwise around the octagon, following edges from point to point, you can see that at every vertex you are making a **left** turn.\n",
    "\n",
    "Now let's consider a non-convex polygon:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c48db38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pacman = octagon[:4] + [Point(0, 0)] + octagon[5:]\n",
    "plot_points(pacman, 'ys-', labels=True, closed=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "The `pacman` polygon is non-convex; you can see that a line from point 3 to point 5 passes *outside* the polygon. You can also see that as you move counterclockwise from 3 to 4 to 5 you turn **right** at 4.  That leads to the idea: **a polygon is convex if there are no right turns** as we go around the polygon counterclockwise.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "# Turn Directions\n",
    "\n",
    "\n",
    "Now how do we determine if a turn from point A to B to C is a left turn at B or a right turn (or straight)? Consider this diagram:\n",
    "\n",
    "<img src=\"http://norvig.com/convexhull.jpg\">\n",
    "\n",
    "It is a left turn at B if angle &beta;  is bigger than angle &alpha;; in other words, if &beta;'s opposite-over-adjacent ratio is bigger than &alpha;'s: \n",
    "\n",
    "    (C.y - B.y) / (C.x - B.x)   >   (B.y - A.y) / (B.x - A.x)\n",
    "    \n",
    "But if we did that computation, we'd need special cases for when each denominator is zero.  So multiply each side by the denominators:\n",
    "\n",
    "    (B.x - A.x) * (C.y - B.y)   >   (B.y - A.y) * (C.x - B.x) \n",
    "    \n",
    "(*Note:* This step should make you very nervous! In general, multiplying both sides of an inequality by a negative number reverses the inequality, and here the denominators might be negative. In this case it works out; basically because we are doing two multiplications so that negatives cancel out, but [the math proof](https://en.wikipedia.org/wiki/Cross_product) is tricky, involving some concepts in vector algebra, so I won't duplicate it here; instead I will provide good test coverage below.)\n",
    "    \n",
    "That leads to the function definition: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "def turn(A, B, C):\n",
    "    \"Is the turn from A->B->C a 'right', 'left', or 'straight' turn?\"\n",
    "    diff = (B.x - A.x) * (C.y - B.y)  -  (B.y - A.y) * (C.x - B.x) \n",
    "    return ('right' if diff < 0 else\n",
    "            'left'  if diff > 0 else\n",
    "            'straight')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "# Sketch of Convex Hull Algorithm\n",
    "\n",
    "\n",
    "Now we have the first part of a strategy to find the convex hull:\n",
    "\n",
    "> *Travel a path along the points in some order. (It is not yet clear exactly what order.) Any point along the way that does not mark a left-hand turn is not part of the hull.*\n",
    "\n",
    "What's a good order? Let's see what happens if we start at the leftmost point and work our way to the rightmost.  We can achieve that ordering by calling the built-in function `sorted` on the points (since points are tuples, `sorted` sorts them lexicographically: first by their first component, `x`, and if there are ties, next by their `y` component). We start with 11 random points, and I will define a function to help plot the partial hull as we go:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "def plot_partial_hull(points, hull_indexes=()):\n",
    "    \"Plot the points, labeled, with a blue line for the points named by indexes.\"\n",
    "    plot_points(points, labels=True)\n",
    "    plot_points([points[i] for i in hull_indexes], 'bs-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "run_control": {}
   },
   "source": [
    "Here are the points without any hull:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXkAAADsCAYAAACR39Z5AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nZuQlqbH/AxfLr4h5SHPkAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10efab828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Now I will start building up the hull by following the points in order from point 0 to 1 to 2 to 3:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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KqufMW3F1+R/Z2vcp4LJ8/fekjc3qoKchn+fML8jNnYC9uv2ZZmNxyBuwKpzeCCzNt/5Z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\nuo+17wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f2a4b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 2, 3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "We see that we made a valid left turn at point 1, but a right turn at 2. So we remove point 2 from the hull:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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OrxyrCs/k1VGl0F9fhtOAcyJ4XMVI6kWDb/V3LvC85qXxHOCGTkXrNp7Jq4oIPgq8twxP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\nS1KLWfKS1GKWvCS12P8DNZrqgpzBy5EAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f3ea908>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "We move on to points 4 and 5:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f5a8160>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 3, 4, 5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Point 4 is a right turn, so we remove it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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bVI5VhWfy6qpS6C8pw5nA+RGsVzGS+tHIW/1dADyteWk8Ffh1t6L1Gs/kVUUEHwDeVoZn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\nFrPkJanFLHlJajFLXpJa7P8BhtYH3p2v9awAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f5a00f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 3, 5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "But now we see point 3 is also a right turn.  The addition of one new point (5) can remove multiple points (4 and 3) from the hull. We remove 3 and move on to 6, 7, and 8:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c4c7c88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 5, 6, 7, 8])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Point 7 is a right turn so we remove 7 and move on to 9:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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BdgWuamZRpTjkrd8iuAc4MjdfR5ootXzBkswWkSfuvTM3r4/gn728bW/ghoiY08trleOQ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\nmEPezKzCHPJmZhX2/wkatt8dvyIjAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f149208>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 5, 6, 8, 9])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Point 8 is a right turn, so we remove 8.  But then 6 and 5 are also right turns, so they too are removed. We proceed on to 10:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ef0eb70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 9, 10])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Now what do we do? We got all the way to the end of our set of 11 points, but we only got half the hull (the lower half).  Well, if looking at all the points in left-to-right order gives us the lower half of the hull, maybe looking at all the points in right-to-left order will give us the upper half.  Let's try. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXkAAADsCAYAAACR39Z5AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f3b69e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [10, 9, 8])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Point 9 is a right turn; remove it and move on:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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3aGbmxoj4PPDeHX9PzTRDXhoei2gCfi5wJHA0sLz3YtuiOWH2y1h7B3BSb1uCiHhSZt7b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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f112358>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [10, 8, 7, 6, 5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Adding 5 reveals 6, and then 7, to be right turns; remove them and move on to 4:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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KCSexHv0rSz4GXNCJ+9aVQ96s3nJsB3gqsH06PjuCZzp031pyyJvVWASPUQ3AQgdCXmJr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\ns4I55M3MCuaQNzMrmEPezKxg/w9i1Wkx2EITkAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f882ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [10, 8, 5, 4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Remove 5 and continue on to 3 and then 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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q3hjyUgWZ3AL8X2meEsGmNetRexnyUj2du/ldgaO7uN4FyjRthrxUzznAw+V8qGIdrRQR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\n5CWpxQx5SWoxQ16SWuz/AS3Za1jJwwanAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f8a6710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [10, 8, 4, 3, 2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Now 3 is a right turn; remove it and continue on to 1 and finally 0:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fae6a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [10, 8, 4, 2, 1 ,0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Adding 0 makes,  1, and then  2 be right turns, so they are removed:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ekopmd40kFcyQl6SCGfKSVDBDXpIKZshLUsEMeUkqmCEvSQUz5CWpYIa8JBXMkJekghny\nklQwQ16SCmbIS1LBDHlJKpghL0kFM+QlqWCGvCQVzJCXpIIZ8pJUMENekgpmyEtSwQx5SSqYIS9J\nBTPkJalghrwkFcyQl6SCGfKSVDBDXpIKZshLUg9ExJUR8ZOI+FlE3BoRsyJfZ0URkjSXRUQA/wis\nyczzge3A9U3WNMKQl6TDtwzYk5mP1a/vBn6zwXoOMuQl6TBl5gvA/Ih4Z/3WbwGnNljSQYa8JPXG\ntcDfRcQQ8Cqwv+F6AJjfdAGSVILMXA9cDhAR7wfObraiii15SepExGIiVhGxePKPY3n9uBD4IvD1\nfpY3FUNekqZTBfs6YC2wboqg/9OI2AJsAO7MzHv7WOGUIjObrkGSZreIVVQBvwDYC1xB5lCzRXXG\nlrwkTW8TsIUq4B8ENjdbTudsyUtSJ6oumvOAzWQON11Opwx5SSqY3TWSVDBDXpIKZshLUsEMeUkq\nmCEvSQUz5CWpYIa8JBXMkJekghnyklQwQ16SCmbIS1LBDHlJKpghL0kFM+QlqWCGvCQVzJCXpIIZ\n8pJUMENekgpmyEtSwQx5SSqYIS9JBTPkJalghrwkFez/AVTDziyj81vMAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fce65c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [10, 8, 4, 0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Let's bring back the lower hull and concatenate it with the upper hull:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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7UQ/9PRl8UNbl1EN/VhUHZUkcAlyYd3eP4KaS9TTLIW9mHZEHZR1I/Rn97QY5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\nIW9m1sMc8mZmPcwhb2bWw/4PSEIQOEpJ6f4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f8f9ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 9, 10] + [10, 8, 4, 0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "That's all there is to the basic idea of the algorithm, but there are a few edge cases to worry about: \n",
    "\n",
    "* **Degenerate polygons**: What happens when there are only 1 or 2 (or zero) points? Such a set of points should be considered convex because there is no way to draw a line segment that goes outside the points.\n",
    "\n",
    "* **Colinear points:** if three or more points are colinear, we should keep only the two \"outside\" ones.  The rationale for not keeping them all is that we want the convex hull to be the minimal possible set of points. We need to keep the outside ones because they mark true corners in the hull. We can achieve this by rejecting a point when it is a \"straight\" turn as well as when it is a \"right\" turn.\n",
    "\n",
    "* **First and last points:** An astute reader might have noticed that our algorithm only rejects the middle point, point B, in the A->B->C turn. That means that the first and last point in sorted order will never be a candidate for rejection, and thus will always end up on the hull.  Is that correct?  Yes it is. The first point is the leftmost point, the one with lowest `x` value (and if there are ties, it is the lowest-leftmost point). That is an extreme corner, so it should always be on the hull. A similar argument holds for the last point in sorted order.\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "# Implementation of Convex Hull Algorithm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "def convex_hull(points):\n",
    "    \"Find the convex hull of a set of points.\"\n",
    "    if len(points) <= 3:\n",
    "        return points\n",
    "    # Find the two half-hulls and append them, but don't repeat first and last points\n",
    "    upper = half_hull(sorted(points))\n",
    "    lower = half_hull(reversed(sorted(points)))\n",
    "    return upper + lower[1:-1]\n",
    "\n",
    "def half_hull(sorted_points):\n",
    "    \"Return the half-hull from following points in sorted order.\"\n",
    "    # Add each point C in order; remove previous point B if A->B-C is not a left turn.\n",
    "    hull = []\n",
    "    for C in sorted_points:\n",
    "        # if A->B->C is not a left turn ...\n",
    "        while len(hull) >= 2 and turn(hull[-2], hull[-1], C) != 'left':\n",
    "            hull.pop() # ... then remove B from hull.\n",
    "        hull.append(C)\n",
    "    return hull"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can try it out on our 11 random points, but it is not easy to tell at a glance whether the answer is correct:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Point(x=0.3253748207631174, y=1.7900592822602743),\n",
       " Point(x=1.3968712854329428, y=0.4086086096198411),\n",
       " Point(x=2.7310024878562857, y=0.05565070635109892),\n",
       " Point(x=2.835445111495586, y=1.375183795456248),\n",
       " Point(x=2.7309330192147097, y=1.4818235191572668),\n",
       " Point(x=1.8813296288048804, y=1.666404092312656)]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "convex_hull(Points(11))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "run_control": {}
   },
   "source": [
    "# Visualization of Results\n",
    "\n",
    "To visualize the results of the algorithm, I'll define a function to call `convex_hull` and plot the results: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "def plot_convex_hull(points):\n",
    "    \"Find the convex hull of these points, and show a plot.\"\n",
    "    hull = convex_hull(points)\n",
    "    plot_points(points)\n",
    "    plot_points(hull, 'bs-', closed=True)\n",
    "    print(len(hull), 'of', len(points), 'points on hull')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6 of 11 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fdcdb70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_convex_hull(Points(11))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the octagon and pacman shapes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "8 of 8 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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BxmNoZMTVZTGdFeR24MdwBRmFoZGhi6wuX6T5psiuLq12BXkvzX8vV5CRGRoZc3VZnytI\nGoZGIVxdGq4g6RkahZnq6uIKkg9Do1BTWV1cQfJjaFSgttXFFSRvhkZFSl9dXEHKYGhUqLTVxRWk\nLIZG5XJdXVxBymVoTMQGVpc7gfvHXl1C4DKa14K4ghTK0JiYVKtLZwW5FdjWucsVpDCGxoSNvbq4\ngtTJ0NDgq4srSN0MDZ216OriCjINhobWtP7q8suPwtdPwMlvNO968RVw1Xa4ehvc0Q0aV5AKGRpa\n19qry772x2pn3+8KUrFNqQ+gvMXIiRj5A+A1wA8Bn73wRz/9v8D3xsgbY+QeA6NOhoY2JEZijPxN\njLwL/v1La3/U4f/wmkX9XE8k9eKkIakXQ0NSL4aGpF4MDUm9GBqSejE0JPViaEjqxdCQ1IuhIakX\nQ0NSL4aGpF4MDUm9GBqSejE0JPViaEjqxdCQ1IuhIakXQ0NSL4aGpF7+H6TgTyboqlDhAAAAAElF\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e9125c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_convex_hull(octagon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7 of 8 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPMAAAEACAYAAABmh0A8AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fde8fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_convex_hull(pacman)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "How about 100 random points?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "15 of 100 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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yBDOX5hDMHxE6EMP9AfghcDvwH+48Hr6/XXYNlK3ulIJ8hJKCR+cCXwHGJ4e/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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fb6eda0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_convex_hull(Points(100))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Will 10,000 points be slow? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10 loops, best of 3: 48.3 ms per loop\n"
     ]
    }
   ],
   "source": [
    "P10K = Points(10000)\n",
    "\n",
    "%timeit convex_hull(P10K)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "No problem! Still well under a second! Here's what it looks like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "27 of 10000 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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1ygMYNOdMU9PLSwrwcs39jai1YmUmF2WsalJrZIfixlpmmhe1S66BMqLMSvFr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\nOzOCJhGrEGA0D+GqNXvmIoZNrGNnWJ1r6Iz0cIbtlpoy1iPj/bvRMk9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lsCNQJ6l5ueUx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\n1csnk1eC9037RNJ3Cp3d6MxF9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zkTNTT1DGpm72tF7V+FJ6sd8n0unA4Jw/HKqjp3fCcj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\nNOl7Niali9axff5+6Xhra6+G4g7Mxt5Pd0O4M7mu3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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f157940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_convex_hull(P10K)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "How about a non-random set? Here is a set of coordinates of 80 US cities:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9 of 80 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f8b1da0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P = Point\n",
    "\n",
    "USA = {\n",
    " P(-621, 289), P(-614, 297), P(-613, 319), P(-613, 342), P(-612, 263), \n",
    " P(-612, 332), P(-603, 247), P(-599, 277), P(-592, 238), P(-591, 323), \n",
    " P(-586, 229), P(-581, 289), P(-581, 305), P(-576, 253), P(-568, 260), \n",
    " P(-563, 322), P(-560, 234), P(-560, 285), P(-559, 292), P(-558, 246),\n",
    " P(-557, 259), P(-555, 225), P(-549, 271), P(-543, 321), P(-535, 313), \n",
    " P(-530, 249), P(-524, 278), P(-524, 288), P(-515, 308), P(-505, 206), \n",
    " P(-504, 327), P(-492, 207), P(-488, 194), P(-488, 248), P(-487, 264), \n",
    " P(-484, 305), P(-484, 328), P(-482, 297), P(-480, 289), P(-477, 210), \n",
    " P(-470, 319), P(-468, 291), P(-462, 247), P(-461, 328), P(-452, 271), \n",
    " P(-450, 210), P(-450, 226), P(-450, 245), P(-441, 311), P(-440, 301), \n",
    " P(-438, 233), P(-438, 293), P(-431, 278), P(-425, 266), P(-423, 273),\n",
    " P(-422, 213), P(-422, 236), P(-420, 251), P(-415, 297), P(-413, 196), \n",
    " P(-409, 214), P(-409, 290), P(-401, 181), P(-401, 253), P(-400, 230), \n",
    " P(-400, 282), P(-394, 251), P(-394, 301), P(-387, 263), P(-385, 272), \n",
    " P(-371, 285), P(-370, 285), P(-369, 299), P(-363, 309), P(-357, 292), \n",
    " P(-355, 297), P(-352, 306), P(-344, 314), P(-340, 328), P(-608, 270)\n",
    " }\n",
    "\n",
    "plot_convex_hull(USA)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "run_control": {}
   },
   "source": [
    "A decidedly non-random set of points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4 of 100 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQQAAAEACAYAAABVmQgcAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\npNcdqg1h46UAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10efac7f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "grid = {Point(x+0.5, y+0.5) \n",
    "        for x in range(10) for y in range(10)}\n",
    "\n",
    "plot_convex_hull(grid)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "run_control": {}
   },
   "source": [
    "A variant with some noise thrown in:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "13 of 100 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f13b320>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def noisy(points, d=0.3, seed=42): \n",
    "    \"Add some uniform noise to each of the points.\"\n",
    "    random.seed(seed)\n",
    "    def noise(): return random.uniform(-d, +d)\n",
    "    return {Point(x + noise(), y + noise())\n",
    "            for (x, y) in points}\n",
    "\n",
    "plot_convex_hull(noisy(grid))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Circles and donuts:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "28 of 790 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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MGI9dfK4uVM7VMV7Oqo9OmmggX4wmuiHHMX3rp/qNLL/0yEy/rhKrzRlU+mCU\nX6+Zfmuq96DYR2r6u1SqyB5THDuSnoOdTaPs9/Vwto9ErV7b7QlzWsaQ33dnlR3kBIWJ/L6Xs+qj\nkyYayN3RRH9otTHu+nnZW69M163Sr91Cxct140Xag6efY0ysZUyr7H6d+m/1YnuM30PpuxXXfHWh\nrypVp7zwktpHDRUaZDphRUAB+YiPNWpu8esgFab9x9zyg1YbU/Wd20ZXlTms+t4vg3K7J6debQ6/\nb3Uzg2WTuNTH2Xf1d1OpubZqCXJXRArf97FGzW6AK1LIOQoTWVixoO0YNesRi18GZaPWhM9nDa3t\nVwkrm5PNoXTZ87p00hTTcmOgtVVLkH+PSOF+H/Non4jMJvmpaJJbQaZVbEC3btguGFk45mKBs8RE\nrUkTa1pszhsNF0gTF2P5m3ddo3n3c8qvuU0Bqdrq3Aye+VvlJT2r7ny6J4wipJBfRhO8sWQx/Rj7\n6hNDuwSa16NNvTX5MgLpvlQDp9+LsfzNvcptW2Q0rk8nPl8UYV/LRFep37wP1zuAF3+D378vZiKH\nsfqMunPqnjCyCHH+DWHtg0sFLrk/zOSdf7lmQdozsBVvSvuMzLSSfX6dirwfal9bJX2zn/2c6hy2\nhNFVeZyq3L71De2m6+e2f4slifWYkKJC2k1It8k8tq3h4LhkoryUn32ibt/NErwLQikajltp6n/X\nFxHVY2S2BJMfMx8jkX6+6u2t2lb0wVvmeNkftuTtXF1ZvtimYhLn4lqoSu17W+7QF6uDZfapcibi\nQ1KpmMc4wcRMnhIQmcqmf6nbd7OHzAUhxJyJ6InM7FD64Pj1CNTt4JkZZFXCjtWU5ZK+DHxEVDUn\n6becMVUQaeFBrv5t9n5jfZSwfh39S6VlDE6vDlbZp5Ypv8mqX3omU1/S2/7C2Iff3hEt9y1118jP\nQntstZiIzaFs255RRABNqEP5eY1I8mYblbTNw5/to85Bzh/UUWu8fEmlxftUfHWGXh10D4LTMREf\n9BqNO5UTr4B3PRiaDP5mPDQZaMwGBq25Q+PYEntI3P5OZvGq240Wzcbw1qY9Q0cATbxB8/P6qPK3\nrvlkYHWkskUC72xtX+ruk14dzD5jq67Gkso9EnvSPEYNz+JVt9eW+JXWzubYIJSb2KVyCqdOGB0y\nG8Nb1QH2a5GvYm7+7DrpeakSxxbNv8VidVfNlrE2vU9J/8Uqnrp/2b3U2TeqcC6KLvZnf1t6CqdO\n7FJMZE+6JD4IAAAgAElEQVReLRfzvGRTq4jIxvCmO8C+VR0XAqiv/2bzfi5UDsKIpC8Fj20lg8RI\nTC/2qt4n07XM79NILTooUk2q0g+q6XWR815F/S5kYQEDkZ2Xiczn5XpPgR0R2bzdmrJV+EuVN+9D\nt3aqnj8R/b89V7kPCcJMfbVjBGnJoh4d5JnIWWKSfmAjUTvQxCks3LUkkUiVKU5Oaiqjs2gTfYrN\n1eN7038L55w3ZJrXqPBz+MtsD3Xc3UWHzSwgz6Z/u9+rnhg12VCkLP3ARqK2xGkWr961bCILWahf\nMN+GSZWI87pu254cP7ECRQczH/uw2mgNfa2D3vbgx91tG5Bn27/r79P0Wp61W77+teh9Kide8Txe\nvzl2XkzijbftlN6ZEzhzYiELY3uIHcfOf2+jD1dFdm6RoiI6PpsvJlJu2DMPWNOvQ926Iqqk50+F\nLJufCT3kXyb+9tonU3LHYcHf8+dj8Yv6Bk46rc6c/B+Ami2e2FoOisXs1dYcO09MJnEjVfp11pXX\nnFTi77BWBS6VlVLIHxyfkl92/PQFX5sE5tQ6vMUSSjU9pJ+pKpjUjnrrn75G3smnnwSRYbZOPM7e\ntebQ/YSyCBFi9Th7S7R5+dwC88WytZlUuXznSxt3rPo6rCZ5Gvqx7YsX1ZtzkQril1GbVaLLqnpF\nQWbtqrcN0NYfccU4iAwx9kjdPrufVBYhQqyeZZqb3pheLLWYr51lvmITvLyN6+Ji1kc+xLrsLdpV\ndnTeXqCL1/Bh0NXvXbHUmTXcZtfILER/EKSXCPez+S8Bkd3ZcE/dPruZSBlChFiNE7i/idJvinsk\ndK3V8QBkDZPxAR/pnCjM8M8GQFVJGs0zyXJcRzI4jIhfQ3OVGpe1oZQVgc5eyVGlCvqWrpw8WvO4\neSJSZ35aF4fuiTyLECJDjI2KSR5CuNkXS1Za0VXyMgn0Md34HVWkNS8u7UdlcX37qji4qGVueJoz\nqvD5bEBfXH/VxDjfRJ6Umc0GZkxl0wPR8fh6XTy6J+osQojM4okx0WdNxlbzOQKbJTHEZat1ZYN6\nFhV8VhUToW582j4zCEWR3AnPf6p5fgz7QC99f80xkQTXsyUfGFbOAPNSm95e1oSEp49srXxZgqyP\njsL/rotH90Sdn5wczBrJHdy81fwBhSGIwIWZBY4XsmiDRUzqjCYHriwNvH2x353o4rd6HO7un4mk\nmdRqL4xWZ9vxg2tMU6sVJiICvxETz0xaYjIxzruHrevxzwZF6l+WMGMdB5yhHJ3iQugWrXvCziKE\nyGGs3pzbiPIqXJulOKK1fqCPLg08238T6+HTEJfMv/Ytagbrpb7R04FsPtSbOmuVZgDZN/k6haaK\nU/vzL7M0Q6i2uZi4mG1tHLqLrkTUl2XEjNdy0Paw91k8cUndPfdP+HURQmQS236V24g0E1Ct5vcL\nzPFCiPrftC9x+La5pImr1i1qCn5FBsSifJFmI3+L+s3il0hhqnSRNeDOkUR6NfHM2OUcmVVg85kz\nlT4bUfGnX/Ki7TzmNXz7fXXXv/nDYItQOLkbShZIbzVvqrU5linB1SOu8lvUzG0BOlG/WPUok+r8\nrlXWwBkzi7wUln+Tl9NYeu52El3VbXf+ggyLbSLR3lzF6dslkcO555V117+dA2GDUDi5/K13u1pr\nxhBnEwdjn6peNk6Zfcn/WmUNnKoXxZcUFt9CaBODFK+rLqE0Ow+3EIIKFfw1fPujik3kqLrr3/1h\nySIUTuw7XePhtfmwA/i0j5SPaeoCtmNwZZ6uZtYqexjNpDAbnGwkVFMpI+nTzbNlwKxB3q0wkb3r\n0kxzxOiKUDixr3WNh7fmV8+t24ep4bGaQdiqeCb9+rKZJPaPhFm4qqR1cEobce0Yr6tqY8CsQWJJ\nZAxkqC6N+z0wHlo0uS90jYe35qcGRL0+bA9CUzagqn491csQn3Ew7odZZ+x1r2ljN672dyBfjM5Z\n7bwZEWGIwYRNRk8FwQyCYAFBMKPWaEk/I176S8NyYCWwFbgTWOHQxxpgLPr/OLDW8vdzgWOBycAx\nwHGlT4tsQORmRDZYjlO+J9X9+lireYTzHY7+PdGhDx845dfcZl3DZ04GFgInG++F2e9mR//+1qjP\n6jE9vmk8NLhA4A2b4dyn4XfudOD0dvp0up9mro+s+2b3J4nUN9KWrXHeeDhHqm61879W5tGtdipe\nM6EBbdm6lAZyUySJXOOlv7YQt5igJO3cp0s2qcqtZ0q02eCcuL860ZX+iCLR79OeANtxfKgoVWpR\nfi03S2Lgc68LY49jdXSrL/tLOS4jEnqGiudt4rFpgLZA7om26N98zLPZDXWboCkTqXLrmequI5IU\nwxmvfHNUb7LPALFsdKR6U137yX9mrl015DrbqlMN/OBp4sq2k+7cmLaNq7x8fTzaekCejob8jI/1\nHlSbSDVkdb9Qb12Dve56KKH+DKHd4e1RfzjYR+xsD3b9HQVsiuZdPo4vW1Ee1lC2xiFuC4F7gG0F\nvz8kh2sTYGZ7MLd1hOt4A3AdcIPhuprQwnLC9YwhXJ/i/fNi6wkCdgNmRn/urDYRESNJRM/17QoZ\nFemurm/6Mj24nr0mG3pdNU7ToeXlaxw+G99rs1VCtWbwEhVNVTwXm5S5TSQbxTpSuM6eMplB9lfO\n2Nt8rGP3G5mfpJ6J6A6iXSBPddBQHUNmEWHWjzUov3nNZS1sm/tBcrsgvK1mwtyrwtVt967sOX0t\nHDNbT0UDOV45Y+f6WMPuNzE3yQskbK8fTXlnyg6iCde3Ocg+vRlhf80c7DZwb7rfevjUMzKa2S3c\npVx3nIpT+T0Yx0FeoTCRhT5w7o4I9JNcHk3w1szi2udrpL+3S/7y4c1I99XeAfSPuxp16Rr16dtj\nlU2wWyC2uSb2hZubfQEkY7pJPgYN5A0KEznGS5+NL4j9JD+iTPKAAsKxX9hEFEznE7Tpo/d5sNtq\nPuwrTdho9JXhN4mN98JOgnWlOzf6qksvmrFB/kw5X8/xQSfdE2oWIWSe1vDj/ibM5xO05SZtk1H5\nx739kH0zG4V6sFdJ2oYgYmN4TIzA+oA4F7pri74sxwb5n9EyjfvImxEZTCYyBPJQNNGraxFbMdEt\nVX7brJjaJSHVw9ktacxs7fUqh73dKpvtKo5MpEg12nHsYBZjg/yfaJke8zVe90RbhBTy+WiiW0H2\nrEVseaKrdpO2sJka/BZk8LPX8+vha580Zi45VKe310t2U13K5t6Lpi7N6tIQXTI2yHeis7XC13jt\nTMoWKeRMRTpdYrDxdULUm66Rapo/kX0buuXz1NPDbV3l5UzBtv+6B6+e2pFVjepLDyo+bau1mrUA\n+Vl0rq71NVbzk3FBCpkCsiGabL62SJdc3n4zy/Mnwmeyh0utxCViStS29+q4rGua4dkdOvP+2/UA\nJZLM2WJTrcx+Xbuwj4wIXDyTE74K510LH9gElwr8j8fCv+dfXneMdibighjyrej8PAUy2RuxtdlM\niafYbhD/vcmIqMM+iuML7HG2qflh54psYt/qHtL07+0rn1X33419RMkLO5lTx9PvpLidd23dcQY5\nd+Y/o3/3Is5lUaFOzYv2wCyXJl8D4mHl78MxqykxlzAPKIa1uNTjsK/5sdAQP9P+XaBuzpIuR8kX\n+KiT4gJnA9MAhpDGzvogM5H/R1iAB+DVXSKihepEN3PiyR6u5O+HDQ+dOtY64HcbYbBFDM+FKfhN\nEqx7SJs75OH85gJnYltgKNuP/Xp9HxgFmCCYsB7TFFoRqxwbyE8jsWsNSFAggrZtqMp6T0xVlbau\ntqhnE2mrNWEjqOtFMt0nt9ACHzVjXXOvRgQunMLrbtoV1RlIVJpDgBds/9QtNdsdisczV1Vs3tT1\n3tCHRs1XKYKmwHfJhPJ1NqEXk32ypztf83TvJ5QUv7SFPac6jl0JOwoTgbRK458Is5A+zEXj+ReB\n6zPHcpzKGFRzNUjs8fQPXR1mX/Ms7sdqz/aZAZcRembecBMsuS5s61Y54pRA56JtRQNZFolevywQ\n7+pGUppGTo4UjudbVfFX7VxXOqAqC7o9NahdNc9PSIBLP0XzdFHF9cGSlXsGsi/Ilugcfd33+ja7\neT4QRD6s6HAHeSHCqg0oOszpqMtmbDF+gq10jFHPoLoM0W6r+WJadfvxZycx3jOQ9yln6GW+17b7\nza1CEDlBWYB3eCIot8jJNt7Y9YKtyhijnkHtSMF7O3rzG21duWcgkyLHhERSfeCMu6Z1v6hVCCIB\nyAPRIvzI00a6RU4O2htblTzMr78sZlCDHLwX4mZ39cSgNnPaq5Z2DfYM5GzlJfwnTcyp+0U1QRL5\nnJKQN9PjZmZ1TJsU9Pbe2EW4mdptdvQWztP8InBX13+dkAHb31Yz83Jp12I8kB9EZ2c9SCM00T2R\nmCCJnK5w0/Nrbbh+U21T0NshUh1u5XabwWQgbgbFBZJUpJNozmWJgfbqZr04jGzIfD1pyVzVNjGo\nHg4yES3dPzW1r90TlgmSnPRF+Oux0D31nsfmsWjjKZw6cT5zHvdyYJpWU+oRaTFufj0Ovhlyvr96\nB9xMEvFzCbbd/qd/OxExPPdarFX7amdQ/Qfl5XucV5pWx2mqY69Ict61RdF2p7BwwsuBb7Cmpe3G\nGxNVYic4y/nt59tQXNZf/TVYJEWJcWm7kBtjrcOQk99uk+RKB5E6F3WZqTtVBtVpIE9E6FzrlZ6z\nYzXZuTckNUxkHos2eib8VRFDaUpd8lMnI42z+/3BviWwcjeyf3tSEdNS18pmH6sPblXI/CKBexQC\nrRPnU25ENjOoXqCclXxNHo+tsY69IqlhIsOcc4MHQswS/lmS5J+Yi6VmhOYnzsTX/cG+D3a1KN5e\ncJ4vKcvOXlZPorU1Ipc0kF9EpPEwyG5e1ls3VpOde0NSw0R8JA9lCH+ZhBdPq4NUi6XlYnyRuF2v\nnmcaZ7N6I+V91St/2GR/5uvgT31Kj2HXjw2jzK6FjRG5pGViqz5Ue52rxmt6AC9INslE0huvXgou\nFoRTZfyMmUb20vG6VcjKb5ere2DbtJnU3zvz4DrzfheIqevcjdkWqWK1JRGQL0XkOwZyoJczUjZe\n0wN4QZL5l4eM5Lxr4T2PhF6aSwXO+IbXsdKEZ35/rN74mWUui5Tn/NbzNCFS+z7qv83TkphpQJxP\nScXFHT8iaQavZ9Sua13+4ik2Ihs0kFkgoxET+bZXmtKN2cYgXhFGnq8ICp+uIKBiYqz+Tn3Dm6kc\nRQRbxFzS/TcXHKaPI7F5W45IVF4v+tfObZk+XMukqoZpXcZnOr+0lFEUxGdXZtL9nmLv+z+Tp9Q8\nmdO80pSmNT5AI0gjV0eL9CzI3iUbVC9j1c/b3E+oubu4HBPpiPVc0odjm9jeRF8cQ5GuYWorqdTd\nq/Rzec+W7kJtu7X2Yz+ybKs5bM9DuG8ziBzKvZtXc1j+upUGWuMDNII0cprCbT9QQbxZq705obq/\nYVyNpWXSkWugVtFt8zZzcdfR03hP5MYuNjS7xmuUqQeqYfviDJNIng+/XyyJtLRKYI7Rfla/LPyF\nDGjaBXz5knhKn+DdY1ZMuEZrfIBGkA6T8u6IFuwRkCka4i2y2le/NWyNavm+/YZd+7NNqPYYU9e1\n3UXoxeMukiI1xmfovl51VNWpZZJIIFnP1kjm2UXiIr35ognzvrczp8ls+R6ITONZeYzn/LppprX9\nPLYxSCOII3+ovEzeolngtE0jzRxME6DKjWrp37od9mIDrOr686M/u9RgrWsXye+HOUN3U+HUoLys\nOqYywwtTe1vM0Hwwbz+u5go6PYMfHE14v64cwn3/2RYDEdmxmchuJCUCVhDXSciLr0VvorJYDh96\nuYtuHP/uHkmug1Tf2vX1ZztVLl6LrFu6CS+SXSU2t/1YLmEMUJVRt0iScWfedpKvs/o2l2VXKC/V\nFzd17opaawM1gjxvvTlx9164bJhzbpjHoo0X8bxxycdlZN9E/vXypL86bkU1B8PvoTV/82dFe9VG\n4CaJ2OPq5w2eVqfKo5D10lJd5l1lL6mjAm8ZZcodAeOPRgzk5lb2R2mtDuYdec7/mcJ9t7eFLIwJ\nT43LUA9DM3p5neY7G1Q/ju6gxAbF+KDpUgH8u6P1ePpxgZoYXbui4/qS7/x9eeQChf7f1PYculk4\nX8iXZ/dm4zLS/09vRHPxGnYEkRW9mz+0ybhZCeh+ZU3qqzRuIrsfj0dRTkvYlknidrbFyw/z8UB/\nINdHtP8EyNS2abfVwbwjr2Eis3j17d6Itc2W4NGsHSI9ZpEEFMdPqFnNdVW8bsLn08ziHoklu3CN\nJ5R5m8XA2M4lkfJqZeXqWib48u+7oNtBv3fGCZ5it/XYXFU4KPf6xnjALbjcV+J2d4x6p8ly4N3A\nMOHdKocAh5C/OtN2nXzfE5Tt7wKCYLFm3vOi54cJ7zU+xvPY+rmE+NwI/DhqNzpfnKWHt8W9AJ93\n+H1t2CmZyMBD1WF3ObSuF1+lxzoJ+ApZBpZc4LUCkQ0OzMr3ZVVqfxPApyk7pGk4NnrmlqifbdG/\ntzqMXTWXuYSMJoja0cCJvi4JCwL2BP4w+vMqEe6t26cTdCH++GrpxLx3PxR7avblzOs6V030Yqtq\nwPRn7/AZjxCqMBdlVJms98o1ejaJ26m/lhdJVep8Wp0ZTeHt7kkz+10ydhzxu9zn3oO8XVFlzuqK\nrjsZtJGJIAfBxBYQWcyPJ3xsktdWbMDcaqyLh32MSBi2nffY+AtIyzKNrH3mImdm1YxtpDosP2E4\n7diZ8mPHWbnebF1R1PbyiIHcBzKpK9ruZNCm2hxWXRlz5ms41amgS2Mtb8C0y0cxiRxtJiBNdZMX\nx9GYeiuaiN5MH1KTZMpmvHAma5DHoVwqS8ftpJ4DOUWRQt7bJW13NnAT7bX8x5wpjE6AyAncsvFx\n9h5ESWSLhO5bu3yUUAIRpV3YAp7FYn/ebW6bFW1/kH24VZvywtmvgVoKQpcvpfaZyzYG+beIFLaA\nzO6StjsbuKk2nY2fUs7aK7rGR0NAc8S2FqffHBaTN2b5YUv6sRPRXQ6yDzWoycCy6lqv+XGr75dR\nvxf1OZD9QLZFH3+1a7rudPBGJoTMBtkYhwCj3j06GBGK6oGwi0gNGcmFNRlIfZtEup/iSGC3Pov3\npq4a5NsWo+8/Gw2tH7dKKkt/n6qjC/JBhbcs6IyWoxaISCdeoSYhCN5xO8x+UfjXA7+G9U8OMzHp\nXO445hus3pPQRecS76AOErs9l1v1EwQLCN2wkwndhAsJY0OaB9uxdXPM93MmsInYBWyPV+yePpai\nvUm+P4bQrWq3d22seYjjicB0YJTQhTy3dNzwN8ehW7fk+7WEMTsrAmQUuA84EFgKvDjkOR1C11ys\niabLqTmFU/0kttV5szVt4PM1dp23qD1e1ZKGjRqUlWqavpwsGTPrKaqXzFlI2/IahaYvbo12ynDq\nGoFGJlV+2VVzCV0JMdW3OTTVzGMcqnR2P3NIx86YMrcqj4aqasV1Xe3VR3uayMeseN5rkB9FQzwD\nsnvr9FPQhjsVg1qG25n1K+AvqCd2zwOmAXcBRxGK12sjkXkNcBWxWB4ExWJ3+Fk7KozL2OE8p5Oe\nYzoyM4xcXQHMJQjsVLr0OLEacxehWnSnts+s2lO8vmpY+lzgu4SR2QFxGD88XIJPkfpmorou3457\nCHeR0FmtvQ6CBZfDQUfC1Glw2Lzw06c3wC8+CT9/a52+vUDXXKyJpr+n5k231njTFEUfxiX01ASv\n9gOafLb8m7w4/sKPx0QXk+JeJrI4qG+iEk/dfOzdt87XPdjTs6d7l2q2XSx35uAXBgGH5z42ywXJ\n5kEcSWhMPJZ0gtdasnkVbolxzUMxXuqb/ChgE8VvXx9Jddk8lKCiz+q8lSQX6PTo+Tix8AzKDbK6\n+WQ/1+e+hIl010St22TOFmEnVWfWrYIlyt+zD4TZh8Puk4EfBwEvE4INhASyBhMVRCeuhuqNCu8C\nnkBNVqsWwdsHPV7xQY09IboEszVRO6TiOT2E63MysYciBP3Y2ed16xh+fg1BcFLls+n5jBEyi3HC\nlwGk1+Nu4FOEiXQt7uWUqc2PUQO6FoXaaiAficXAIcZWRNWwi6p4XVQh8qbF1UTNicV//5Xam2jV\nxmHTG9/8GCvTId7tG53N1qO9Oi8J3e4PfzU6yOpM5wi0NtEwYenz8QYs4MaJjUyPiSF2/+UvMzJp\nZYeuS5duNc52eLlGqZrjYmK3aCrqVF2PYsbYhqs4TbN7h0l2lxYwEBkYJrJTBpvpIAiYBHydSNc5\nnavlu7zm11PZcgZwFvA5mghIqgoq6gps8Mp7UiDx3NQN3KsOBqsKSPMBQTAS4XFoboz0+GsiHIu9\nPF5QYXfgJ8B8+Bjw2MPwwKr0U+tWifTemdYb4VUTP4y5+W5s/ibI0MBKDIPSij0pftQOk7VvQy30\ndXOi2/y3S1ggU1QajZLtOkv1r2qdI9DJpJE9oryaeJM+CxKUqiW+2yDk8djj23QqvYkdpn5Amssc\nm5p/RpW7neNnglyp0OZVILt1vv8lrXMEOps4sg/hpVfxZv1t4+OmjYfNJYQ1i3/VQW+OMZow+brx\nK9X2rbi4kPeKdOMEWw7l3u8qNHnjoESllrXOEeh08sgBIGuUTbuksfHyHo3B89jUn5/bFQzFfbkx\no+bVjnIGZYu7IuG8g888qtDiMpBZne+rQescga4byBEg6ub9QSNv1Dxxt2blL8Clifm5XcFQjFtd\nSaIZtcssnyhbj9Ykj2rxSdzweYUGfwPSzk2DHlrnCAxCA3lRlNAkMDH2byy517uqkSfu5uMhipiF\nj3D14rF8MZH6kkRTtq0qBlX8oqiSWpZezkVjCgN5COSwxmiigdY5AoPSCGtWjoLIVDbJ9bxMnIlY\n19oy3KYzY7O5IM2I+1VBd3b91JMkkvnXt11kGbF5TFC1ygoLvsk524YYExAZZuszIHMbpY0G2i4V\nJ1IFQcDvgXwHgkl78gxXc8aqBdx8AoMU21EFSTzDccAkwnyUJPYiXeDnbsLLqm7xMkdf8TCm/RRl\n1yYXRsXpCcuBk5zwcYlNSRcSuopMIaUkIxeG2GOfCQ6cC0MMMSxzOO8Vd8tR/22NZ9fQNRcbtAby\npli0jG5an2PVR9eu23xtznxF+cTLUP8OlK7mq8+6La7rUX8tXcsypqQWXUbuEG/+dSf04qHtYlm8\n1SDCvxK+nRGG9iVM2Nt/+wNlGbm6W+jazeJdThhRGcM48C7Sb+q5JDeyuWfhut665wd0WbdxoqRE\nLU6UdIF6N/dZXI85wcYnnDAcANhJs3jrgQifCgL2AT4IHzsU1q8KgvuWDrNNXsyiF01lYvr+PPDk\nN4LgeZnDeT4JYR9HmDZ+K21m8YaZrgsJD/YhhMR/q4KjGrquLzpkBkUH2b5mqxsUZxuH8z+JsN4p\nwK2FY5ngYpo1bNh3EDAE+x1q1MeOBF2LQoPaooS9f9YlP53Cwont4m1atB6VdJ3N1jM/FZzSBkDf\noev2UaRp9aOuKuRqqG7KQ1XSN8gMkO8OejKdS+vVGQ2E9MA7YeNvi77fzKSwunkI6ht5mPCS6YDw\nLS/4vczaDIpF6ax4fmvBM3rIqmV2F4/ni/vUVYUs1IUKXNxUuWIVNdd3EDCHUEJ7tfU4OwD06kwJ\niDAeBOvuBGYnn34M2Mwt7D0WcN73CJYwzDmTzuWO9d9g9QxCjwckasKthAfNzdvgd0Ju4nmCm04t\nCwx6yKof0yM8hjFRhfzCckJVLvZQ2TH28rVIzfMQ1jyX0EuzV/j15vvh9ffD+Hi603WZDN0dCLoW\nhQa95a3plxaKo8Occ8N20ToRs9MRi/rgL/N8mu68IXlPha1akF6X6ou4/eKfrFvdEH2DyNWtDM+f\nyVPvBxlX6OSjDHA2rmvrHIFBb6ZMJKfT5g9YfHVm1iZQnk+TJ361iLK/RLCqVlSQxywMPM/w0r/b\nKq7Rrea4Z/fC3k6VDmArvU8GZBrIvyr0sQnk/K5puanWqzOVkK7XOgQnToSieBVkdeNrgIOi72Lx\nXS1MfAjZmqX5YKf/Qfo6hKuj55r1+KSL9awBziRUjfT1WC1EfmLvUXOQ3QupwNskgO10olvpSHtg\nDgS+A5wQfXQ/8BoRftXM1AYAuuZiO1ob5pwbiiWR9z4DctT2Z/Uh0BL9PSPzTD6fpvxKBT+3+VW1\nEMdVCu5m9VjbuvzKfA5pyaHYe+UewAYzLuFTb40CFONHrwfZt2uabbp1jsCO1vR3gFwqUe7NX2zX\ne/M2gG0Cq0Wt36mznyTfFRF/HG3afBZwPgJ2ldF4TWbTus3DpBbJYoVZZFVKvQ0HZnySP7t/MltU\nmvgcA15MyFfrc2csQc19CGFoCPY7DJ67P7w/9lL8AniLCCujH8XicCw+p3M5ynI0dHkkPvJUTLxB\n6VybtdjUFvVZW7Zpz1VVzk1yYTcoAWxBwOQ53PP11RxxLsBktvJSbvrYtXLq+73jOKjQNRfbWRrI\nSSCrlDfRFpD3gwwbiPbtXyth46Gozlx1LUVo9rsmg8OK90Bv7FXwBpkN8tN4z/flEfkhr7inc6mr\n5dY5AjtTi6zy/5Bx6936Ym6bVyradyH6+6j/4Xq47V3DbRRpLrJPaWux3MT8uwPG18b7PIltv/oX\n3vSqXY2BiPRMpJlFReaD3Kkwkq17sP7DjzL7JC2RtWloDMfzwUTcDrft71yZrK2UlLZP6QysW6/k\nXJnORlH292sg07qmu65a5wjsrA1kKsjfqVLJJLYt/Rsu+8OBeFsl6ox7EaG8B8qspJ8LU7BlsnVU\nII2BdTWH7XkJn3okYR4TE5EhPeh8PztsnSOwszeQE0GWJ5GtW+XdfOKR/+C1e3eNmxfpJwlCs1Vp\nRiS8srSZWqJu0k4cOZzzxIDMBPmewkCeBjmj8z0cgNY5ArtCA5kyl2VXTFJCDXZnw2qQl3gfr+2w\neEXietgAAAZTSURBVBeVph1DqWuGsRrTs1VgEciRGfX0TpAju6arQWm9i7ctCIIZN/LS297GPx/x\na14Qu4LHgY8DHxJhi2u/xK7PEJq9arJ4/NgFbHalpsm1mf5wK3cxJ3VgPhvj8wccPrqOA/fczKRN\nt3HImgkOPBqGJsFU4H3fB94ownrv+O6o0DUX26UazHiQkZftwfqPgGxT3mwrQOa59Jd5ow9O7RIz\nvLu+MkM1oG6K8ZnC624qDij807UgQ53T0YC1vp5ImyCyYX956GcbZMZfEeZWxPkUxwI/DwI+HgRM\ntehRlxOir13SRKlG27oednVI/EK6pON1JOs3CXj7idxy0RYOOqj4x4/cJ8JES5juMNCrMx1CEDAZ\n+EvgUkJCBi57BtbeC89mxOWCG+CLVIkQikV4l+rlOxvkVam165lxyJe48NG/4B8eHmd4HlxG2LKw\n5DqRK09tDdcdBPos3g5BhG3AR4KA/wS+TCidzIQvvyj/9JL8R0mRoROJCwOFTEFnX7CriaoD83D5\n5sLU3WE5sHKC4Jj/ZvH95/KtX25kjwMnmHQQSZZ1DxbQqzMDACIsBxYA7wt17wrIqySfIiwLcGOF\nmlKvenk8dlVF+26rwKdxzahuATJ9Jk9/awYbHnolP56znpnnTzBpWvT1Q8BH4Te3dILvDgq9JDIg\nIMIY8PEguO8ckkQvBQ48Kgh4vhCsIa2SfJDkCoi50W+v0QziXh4xgbw0EwQr0Nc9abv0YQiK6raN\n4ZXPCx74yIMc+Abg7PXMVOl+DPge8CXghyKMBcHq2bBkNN/pDlzCsEHomcjAweZNxZ/vuR+wbB8e\nv+0K3nLc2Xx/eBITxxAWCjKHcnXHBIoKEZkX/WkP5t7DnGO/zJsnf4ULXvgQB1yZ+f4uQsbxVREe\nVb/I2Z56KIXesDpgEARLroUrF+a/uQzV2Hc4q5nBXzx7J8MrXsiTx01lYvpmhjbdzqxfjfHgnSI/\nf6uTXcLc3pFIM7YGXo8wLZh3xTRGjt/A5GfHGBoPSzPsNRtm7QV/t1/m8WeBfydkHj8Pvb091IVe\nEhk4SJdjTGDDw8DDwIXAnr9hDvDC3eGyeYoCvzvwMlgynilpaHZpVnlJwwSy0oxeTWpGhUkY3Zrj\neMWSX/Kfu+cfumz7/yYx9otxhr8AXCnCIBl5dwromciAQZUoHQRcCvwRcAkwp/AZ9j/y8/zx7S/n\np/sdySoCc7uEu/emvpqUgiAgAPZ+IUuPOI4VL72aM7Y9yT6zp7D50BO5/jVPsM8eD3CgbOAfNc6B\n8W3Ap4ErxmR4pS+8eshDz0R2MIjepJ8JAj4LDywFnp97hr1G3ha9iWfxJC/kjrHbOOH3NgbsBdwi\nwpOa7vWFlz1CEDAJ2Bc4UGkHFPx/6h0czx0cv/23W5jKz7ZrS2X33az6uQh/3gD6PWSgZyI7KIgw\nEQTrNcxgYoLIff8Ue3MtL58OfCD+Ngi4m7CE483w6tNgynMAgfMYZmLDDMZWjPLQ0lG5xVr0DwJ2\nA/annDnsTxghagkiAfLYcazY8zDunbIXTz/5NX67aRwOLnrWvv8eXKBnIjsl3PWz57PsA3vx9Ktv\nYd4BW5j6EsIb+WI4Kmpvgheh2g/GgKf4GPDLw4JgyWHpfh+5F67/OHrmcCChhOECY8CDUXsAeGAm\nTz/+j7znj49k1f778cg9q5nzsjPk6qcIXrDdsPuvnPc9CplID21Bz0R2aNAZYdetWiYvuJGw8DAA\nQcDewDzgd4D50b+zivvdDHxzJmFuiwKXLQTe7IDoKBFjiNqDBX8/ls9L2QuCL/8TEcOYI6tDyUi1\nvwQLtGvggGcPDtAzkR0YbOIZIjvI1VEjCBgCjoBHvgscXQONpylnDg8ATzu7UysMtn1MR/fQM5Fd\nFKK3/t1B8OSjGDORx+8jTBaMmcODIjzbFI497BjQM5EeLOCx+0X4atdY9DBY0Ees9tBDD7Wgz+Lt\noYceakHPRHrooYda0DORHnrooRb0TKSHHnqoBT0T6aGHHmpBz0R66KGHWtAzkR566KEW9Eykhx56\nqAU9E+mhhx5qQc9Eeuihh1rQM5EeeuihFvRMpIceeqgFPRPpoYceakHPRHrooYda0DORHnrooRb0\nTKSHHnqoBT0T6aGHHmpBz0R66KGHWtAzkR566KEW/H9ksqnHR1PVnQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f3d7e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "square = {Point(random.uniform(-1, 1), random.uniform(-1, 1)) for _ in range(1000)}\n",
    "circle = {p for p in square if p.x ** 2 + p.y ** 2 < 1}\n",
    "plot_convex_hull(circle)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "28 of 628 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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d2eMFmXVNH02Ic4Evp5HMQfONwovxuiWmzeRtepEQCK8fs+ZyV6f78f48dQYc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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f5cc978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "donut = {p for p in square if 0.2 < (p.x ** 2 + p.y ** 2) < 1}\n",
    "plot_convex_hull(donut)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "19 of 720 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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8+MFmSvNMo7DSViSMrqMe0j0j0r7LFOACYFveBajHFcD37P+LgO/aJngNQ0N9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\nBv8fLdTLMnE7hQAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10efcd3c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def rad(degrees): return degrees * math.pi / 180.0\n",
    "\n",
    "sine = {Point(rad(d), 5 * math.sin(rad(d))) for d in range(720)}\n",
    "plot_convex_hull(noisy(sine))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "38 of 360 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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mNDerlZNtqWQOf73Dk8/m5pT3sRWt0/gZKMWJmBj3rMm8pd/kVJVl4tLqgWl4Kwf7d2y+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\nYSAIguX/A4KDcggsT5ndAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fcf3400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "donut2 = noisy({Point(5 * math.sin(rad(d)), 5 * math.cos(rad(d))) for d in range(360)})\n",
    "\n",
    "plot_convex_hull(donut2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Tests\n",
    "\n",
    "So far, everything looks good! But I would gain even more confidence if we could pass a test suite:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'tests pass'"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def tests():\n",
    "    # Tests of `turn`\n",
    "    assert turn(octagon[0], octagon[1], octagon[2]) == 'left'\n",
    "    assert turn(octagon[2], octagon[3], octagon[4]) == 'left'\n",
    "    assert turn(octagon[1], octagon[0], octagon[7]) == 'right'\n",
    "    assert turn(octagon[5], octagon[6], octagon[7]) == 'left'\n",
    "    assert turn(octagon[2], octagon[1], octagon[0]) == 'right'\n",
    "    assert turn(pacman[1], pacman[2], pacman[3]) == 'left'\n",
    "    assert turn(pacman[3], pacman[4], pacman[5]) == 'right'\n",
    "    assert turn(Point(0, 0), Point(0, 1), Point(0, 2)) == 'straight'\n",
    "    assert turn(Point(2, 1), Point(3, 1), Point(4, 1)) == 'straight'\n",
    "    assert turn(Point(2, 1), Point(4, 1), Point(3, 1)) == 'straight'\n",
    "    assert turn(Point(0, 0), Point(1, 1), Point(2, 2)) == 'straight'\n",
    "    assert turn(Point(0, 0), Point(-1, -1), Point(2, 2)) == 'straight'\n",
    "    # More tests of `turn`, covering negative denominator\n",
    "    A, B = Point(-2, -2), Point(0, 0)\n",
    "    assert turn(A, B, Point(1, 3)) == 'left'\n",
    "    assert turn(A, B, Point(2, 2)) == 'straight'\n",
    "    assert turn(A, B, Point(3, 1)) == 'right'\n",
    "    assert turn(A, B, Point(-1, 1)) == 'left'\n",
    "    assert turn(A, B, Point(-1, -4)) == 'right'\n",
    "    assert turn(A, B, Point(-1, -1)) == 'straight'\n",
    "    assert turn(B, A, Point(-3, -4)) == 'left'\n",
    "    assert turn(B, A, Point(-4, -3)) == 'right'\n",
    "    assert turn(B, A, Point(-1, -1)) == 'straight'\n",
    "    assert turn(B, A, Point(-3, -3)) == 'straight'\n",
    "    \n",
    "    # Tests of convex_hull\n",
    "    assert convex_hull(octagon)== octagon\n",
    "    assert convex_hull(circle) == convex_hull(donut)\n",
    "    assert convex_hull(circle) == convex_hull(convex_hull(circle))\n",
    "    for n in (0, 1, 2, 3):\n",
    "        assert convex_hull(Points(n)) == Points(n)\n",
    "    collinear = {Point(x, 0) for x in range(100)}\n",
    "    assert convex_hull(collinear) == [min(collinear), max(collinear)]\n",
    "    P = Point(5, 5)\n",
    "    assert convex_hull(collinear | {P}) == [min(collinear), max(collinear), P]\n",
    "    grid1 = {Point(x, y) for x in range(10) for y in range(10)}\n",
    "    assert convex_hull(grid1) == [Point(0, 0), Point(9, 0), Point(9, 9), Point(0, 9)]\n",
    "\n",
    "    return 'tests pass'\n",
    "        \n",
    "tests()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "## How Many Points on the Hull?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "The number of points on the hull for `Points(N)` seems to increase slowly as `N` increases. \n",
    "How slowly? Let's try to find out.  We'll average the number of points on the hull for `Points(N)` over, say, 60 random trials:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "def average_hull_size(N, trials=60):\n",
    "    \"\"\"Compute the average hull size of N random points\n",
    "    (averaged over the given number of random trials).\"\"\"\n",
    "    return sum(len(convex_hull(Points(N, seed=trials+i)))\n",
    "               for i in range(trials)) / trials"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll do this for several values of *N*, taken as powers of 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   N   Hull Size\n",
      "   1:  1.0\n",
      "   2:  2.0\n",
      "   4:  3.7\n",
      "   8:  5.1\n",
      "  16:  7.1\n",
      "  32:  8.6\n",
      "  64: 11.0\n",
      " 128: 12.6\n",
      " 256: 14.6\n",
      " 512: 16.4\n",
      "1024: 18.1\n",
      "2048: 19.8\n",
      "4096: 21.6\n",
      "8192: 23.2\n"
     ]
    }
   ],
   "source": [
    "hull_sizes = [average_hull_size(2**e) \n",
    "              for e in range(14)]\n",
    "\n",
    "print('   N   Hull Size')\n",
    "for e in range(14): \n",
    "    print('{:4}: {:4.1f}'.format(2**e, hull_sizes[e]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then we'll plot the results, with *N* on a log<sub>2</sub> scale:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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58UbVInLxgwVm9l/Ad+7+UNyxJCP64HM9kNgWMqm/4WxMBouAnROud45uyxnR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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fcdacc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_hull_sizes(hull_sizes):\n",
    "    plt.plot(hull_sizes, 'bo-')\n",
    "    plt.ylabel('Hull size')\n",
    "    plt.xlabel('log_2(number of points)')\n",
    "\n",
    "plot_hull_sizes(hull_sizes)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That sure looks like a straight line! \n",
    "\n",
    "That means we can define `estimated_hull_size` by computing a slope and intercept of the line. (I won't bother doing linear regression; I'll just draw a straight line from the first to the last point in `hull_sizes`.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fdf8a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def estimated_hull_size(N):\n",
    "    \"Estimated hull size for N random points, (inter/extra)polating from hull_sizes.\"\n",
    "    slope = (hull_sizes[-1] - hull_sizes[0]) / (len(hull_sizes) - 1)\n",
    "    return hull_sizes[0] + slope * math.log(N, 2)\n",
    "\n",
    "# Plot actual average hull sizes in blue, and estimated hull sizes in red\n",
    "plot_hull_sizes(hull_sizes)\n",
    "plt.plot([estimated_hull_size(2**e) \n",
    "          for e in range(len(hull_sizes))], \n",
    "         'r--');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's an estimate of the number of points on the hull of a quadrillion random points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "86.15634904778565"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "estimated_hull_size(10**15)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "# Concluding Remarks and Further Reading\n",
    "\n",
    "The convex hull problem is an interesting exercise in algorithm design.\n",
    "The algorithm covered here is called  [Andrew's Monotone Chain](https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain).\n",
    "It is a variant of the [Graham Scan](https://en.wikipedia.org/wiki/Graham_scan).\n",
    "You can read more from [Tamassia](http://cs.brown.edu/courses/cs016/docs/old_lectures/ConvexHull-Notes.pdf) or [Wikipedia](https://en.wikipedia.org/wiki/Convex_hull)."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}
